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Question
A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.
Solution
Let 2a and 2b be the sides of rectangular portion ABCD.
Given length of running track = 440 m
πa + πb + πa + πb = 440
`\implies` 2πa + 2πb = 440
`\implies 2 xx 22/7(a + b)` = 440
`\implies` a + b = 70 ...(1)
Area of Rectangle ABCD
A = (2a)(2b)
= 4ab
= 4a(70 – a)
= 4(70a – a2)
∴ `("dA")/("da")` = 4(70 – 2a)
For max/min `("dA")/("da")` = 0
4(70 – 2a) = 0
∴ a = 35
Also `("d"^2"A")/("da"^2)` = – 8a
= – 8 × 35 < 0
So, A is maximum when a = 35
By (1),
b = 70 – a
= 70 – 35
= 35
Hence sides of rectangular portion are
2a = 2 × 35 = 70 m
and 2b = 2 × 35 = 70 m
2nd part: Area of whole field
= `2 xx 1/2 π"a"^2 + 2 xx 1/2 π"b"^2 + 2"a" xx 2"b"`
= πa2 + πb2 + 4ab
= `22/7 xx (35)^2 + 22/7 xx (35)^2 + 4 xx 35 xx 35`
= 3850 + 3850 + 4900
= 12600
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